Задачи по векторному анализу. Михайлов В.К - 98 стр.

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98
èíòåãðàëû ñâîäÿòñÿ ê îáûêíîâåííûì. Ýëåìåíòû dl êîîðäèíàò-
íûõ ëèíèé â öèëèíäðè÷åñêîé è ñôåðè÷åñêîé ñèñòåìàõ ïðèâåäå-
íû â êîíöå ðàçäåëà 4.1.
Ïóñòü â êðèâîëèíåéíûõ êîîðäèíàòàõ (q
1
, q
2
, q
3
) çàäàíî âåê-
òîðíîå ïîëå
ρρρ ρ
aaeae ae=+ +
11 2 2 33
. Òîãäà îðèåíòèðîâàííûé ýëåìåíò
dl
ρ
ïðîèçâîëüíîé êðèâîé L â ýòèõ êîîðäèíàòàõ
dl H dq e H dq e H dq e
ρ
ρρρ
=+ +
111 2 22 3 33
.
Ðàáîòà ïîëÿ
ρ
a
âäîëü êðèâîé L åñòü êðèâîëèíåéíûé èíòåãðàë
ρ
ρ
adl aHdq aHdq aHdq
LL
⋅= + +
∫∫
()
11 1 2 2 2 3 3 3
.
 ÷àñòíîñòè, â öèëèíäðè÷åñêèõ êîîðäèíàòàõ (q
1
=ρ
, q
2
=ϕ
, q
3
=z;
H
1
= 1, H
2
=ρ
, H
3
= 1):
ρ
ρ
adl ad ad adz
LL
z
⋅= + +
∫∫
()
ρϕ
ρρ ϕ
;
â ñôåðè÷åñêèõ êîîðäèíàòàõ (q
1
=r, q
2
=θ
, q
3
=ϕ
; H
1
= 1, H
2
=r,
H
3
=rsin
θ
):
ρ
ρ
adl adr rad r ad
LL
r
⋅= + +
∫∫
(sin)
θϕ
θθϕ
.
Ïðèìåð 1. Âû÷èñëèòü ðàáîòó (öèðêóëÿöèþ) ïîëÿ
ρρ ρ
are r e
r
=+sin
θ
ϕ
ïî îêðóæíîñòè r=R,
θ=π
/2.
Ðåøåíèå.
ρ
a dl a dr r a d
r
⋅= +
∫∫
(sin)
θϕ
ϕ
.
Íî òàê êàê íà îêðóæíîñòè r=R, dr = 0,
θ=π
/2, òî
ρ
ρ
adl R d R
L
⋅=+ =
∫∫
02)2
22
0
2
2
sin ( /
πϕπ
π
.
Ïðèìåð 2. Âû÷èñëèòü ðàáîòó ïîëÿ
ρρρρ
aezee
z
=++4
ρϕ ρ
ρϕ
sin
îò
òî÷êè (0,
π
/4, 0) äî òî÷êè (R,
π
/4, 0) ïî ïðÿìîé L:
ϕ=π
/4,
z =0.
Ðåøåíèå. Íà äàííîé êîîðäèíàòíîé ïðÿìîé z = 0, dz =0,
ϕ=π
/4, d
ϕ
= 0; è òîãäà
èíòåãðàëû ñâîäÿòñÿ ê îáûêíîâåííûì. Ýëåìåíòû dl êîîðäèíàò-
íûõ ëèíèé â öèëèíäðè÷åñêîé è ñôåðè÷åñêîé ñèñòåìàõ ïðèâåäå-
íû â êîíöå ðàçäåëà 4.1.
     Ïóñòü â êðèâîëèíåéíûõ êîîðäèíàòàõ (q1, q2, q3) çàäàíî âåê-
            ρ      ρ     ρ      ρ                                   ρ
òîðíîå ïîëå a = a1e1 + a2e2 + a3e3 . Òîãäà îðèåíòèðîâàííûé ýëåìåíò dl
ïðîèçâîëüíîé êðèâîé L â ýòèõ êîîðäèíàòàõ
                      ρ         ρ          ρ          ρ
                     dl = H1dq1e1 + H2 dq2e2 + H3dq3e3 .
             ρ
Ðàáîòà ïîëÿ a âäîëü êðèâîé L åñòü êðèâîëèíåéíûé èíòåãðàë
                 ρ ρ
               ∫ a ⋅ dl = ∫ (a1H1dq1 + a2 H2dq2 + a3 H3dq3 ) .
                L               L

 ÷àñòíîñòè, â öèëèíäðè÷åñêèõ êîîðäèíàòàõ (q1 = ρ, q2 = ϕ, q3 = z;
H1 = 1, H2 = ρ, H3 = 1):
                     ρ ρ
                   ∫ a ⋅ dl = ∫ (aρ dρ + ρaϕ dϕ + az dz ) ;
                        L           L

â ñôåðè÷åñêèõ êîîðäèíàòàõ (q1 = r, q2 = θ, q3 = ϕ; H1 = 1, H2 = r,
H3 = r sinθ):
                ρ ρ
              ∫ a ⋅ dl = ∫ (ar dr + raθ dθ + r sin θ aϕ dϕ ) .
                L               L

      Ïðèìåð 1. Âû÷èñëèòü ðàáîòó (öèðêóëÿöèþ) ïîëÿ
ρ ρ               ρ
a = rer + r sin θ eϕ ïî îêðóæíîñòè r = R, θ = π/2.
     Ðåøåíèå.
                            ρ ρ
                         ∫ a ⋅ dl = ∫ (ar dr + r sin θ aϕ dϕ ) .
Íî òàê êàê íà îêðóæíîñòè r = R, dr = 0, θ = π/2, òî
                        ρ ρ                         2π

                    ∫ a ⋅ dl   = 0 + R2 sin 2(π / 2) ∫ dϕ = 2πR2 .
                    L                               0
                                      ρ           ρ     ρ     ρ
      Ïðèìåð 2. Âû÷èñëèòü ðàáîòó ïîëÿ a = 4ρ sinϕ eρ + zeϕ + ρez îò
òî÷êè (0, π/4, 0) äî òî÷êè (R, π/4, 0) ïî ïðÿìîé L: ϕ = π/4,
z = 0.
      Ðåøåíèå. Íà äàííîé êîîðäèíàòíîé ïðÿìîé z = 0, dz = 0,
ϕ = π/4, dϕ = 0; è òîãäà




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